Polyomino strips, snakes, and ouroboroi
Previously on this blog: “Polycube snakes and ouroboroi” (20221118).
Preparing to add the sequences from that post into OEIS, I realized to my surprise that most of the 2D analogues weren’t yet in OEIS either! There are several nearmiss variations, though, involving the following concepts:
Polyominoes with holes (snakes versus strips)
Any polyomino divides the plane into one or more rookwiseconnected regions. Two chess rooks are in different regions if one cannot reach the other in any number of rookwise moves without at some point passing through a cell of the polyomino. A polyomino which divides the plane into \(k+1\) regions is said to have \(k\) holes.
The smallest polyomino with a hole is the Oheptomino… which also happens to be a polyomino snake.
The polyomino divides the two pictured rooks from each other; therefore it is not a strip.
Miroslav Vicher refers to polyomino snakes without holes as strip polyominoes. Strip polyominoes are interesting if your main recreational interests in polyominoes are tilings and dissections, because holes pose a problem for those tasks. The 30 free heptominostrips can tile a pretty 14x15 rectangle:
but the 31 heptominosnakes can’t tile any holefree shape at all. Vicher notes that the 150 free nonominostrips cannot tile any rectangle, either, because the nonomino depicted below has — not a hole, but what Vicher calls a cave.
Although the interior of the cave can be reached by a freely moving rook, it cannot be filled by any possible combination of snakes.
Vicher’s website often uses the shorthand “polystrip” to mean “strip polyomino”; but that’s unnecessarily confusing, given the established convention of using the suffix to identify the base form: polyomino, polyiamond, polyabolo, polyhex. We can speak also of strip polyiamonds, strip polyabolos, strip polyhexes; and snakes and ouroboroi of those forms as well.
OEIS sequences demonstrating the effect of holes include:
 OEIS A049429 counts free ddimensional polycubes
 OEIS A000105 counts free polyominoes
 OEIS A000104 counts free polyominoes with no holes
 OEIS A057418 counts free polyominoes with exactly one hole
 OEIS A001419 counts free polyominoes with one or more holes (i.e., A000105 minus A000104)
 OEIS A002013 counts free snake polyominoes
 OEIS A333313 counts free snake polyominoes with no holes (i.e., strip polyominoes)
 OEIS A038119 counts free polycubes
 OEIS A357083 counts free polycubes with one or more holes
(Just to confuse matters, the title of OEIS A151514 and a 2009 comment on OEIS A002013 formerly used the term “strip” when they meant “snake.” But I submitted corrections and these are now fixed.)
Holes in 3D
There are at least two ways to define the 3D equivalent of a “strip” (resp. “a snake without holes”).

The simplest kind of 3D “hole” is a fully enclosed empty region from which a 3D rook could not escape by 3Drookwise moves; this is also known as a “cavity.” The smallest polycube with such a cavity is also a snake:
SRDRURDRUR
wraps around all six faces of an empty cell, in the same way as the Oheptomino wraps around all four sides of an empty 2D cell. Similarly, the shortest polyhypercube snake with a cavity would use 15 hypercubes to wrap around all eight hyperfaces of an empty 4D cell; and so on. 
A different kind of 3D hole is the “donut hole”: a tunnel through which you can pass a string, then connect the ends of the string to form a loop, such that the loop of string cannot escape to infinity no matter how much it wiggles. Of the three free eightcube ouroboroi, one (the simplest) has a donut hole, and the other two don’t. Of the two chiral tencube ouroboroi, one has a donut hole, and the other (being the polycube equivalent of a horn torus) doesn’t.

My lovely wife pointed out the donuthole possibility; I then further observed that some polycube ouroboroi have what I might call “pseudotunnels,” where the ouroboros itself actually does not circumnavigate the tunnel, but simply doubles back on itself. We can characterize these pseudotunnels by whether our loop of string can escape them by wiggling through corners, or whether it must wiggle through edges as well. I believe the smallest polycube ouroboros with an “edgetype pseudotunnel” is this quite appropriately shaped 22cube construction:
A minor tweak gives a 24cube ouroboros with a “cornertype pseudotunnel”; this can be reduced to a 22cube ouroboros (see my followup post).
Onesided versus free polyominoes
Whenever we talk about the distinctness of ddimensional shapes, we must clarify what kinds of rotations we’re permitting.
 No rotations at all: “fixed” polyominoes
 Rotations in dspace but no higher: “onesided” polyominoes
 Rotations through d+1space: “twosided” or “free” polyominoes
When counting “onesided” 2D polyominoes, we don’t allow flipping them over through 3space: mirror images are counted as distinct. Likewise, when counting “onesided” 3D polycubes, we don’t allow flipping them through hyperspace: mirror images are counted as distinct. A polyform with distinct left and righthanded variants is called “chiral”: its left and righthanded variants are counted as two distinct onesided polyforms, but identified as the same free polyform.
This notion of onesidedness is different from my previous blog post’s notion of “directed snakes,” where I was distinguishing the snake’s head from its tail.
The mathematical term for that concept seems to be “rootedness,” although you might need some more verbiage to explain that a “rooted snake” must have its root actually at one of its ends, and not somewhere in the middle.
The following image shows just four free undirected polyomino snakes (the I, L, W, and Z pentominoes); but since the L and Z are chiral, we have a total of six onesided undirected snakes in the left half of the image. Meanwhile, variants of the L and W produce six free directed/rooted snakes across the top half of the image; and the entire image shows nine examples of onesided directed snakes. (Of course this is just a representative sampling of the total number of onesided directed snake pentominoes.)
OEIS sequences demonstrating the effect of chirality include:
 OEIS A000988 counts onesided polyominoes, which is the sum of:
 OEIS A000105 counts free polyominoes
 OEIS A030228 counts chiral polyominoes (i.e., A000988 minus A000105)
 OEIS A000162 counts onesided polycubes (A355966 counts those with cavities)
 OEIS A038119 counts free polycubes (A357083 counts those with cavities)
 OEIS A068870 counts onesided polyhypercubes
 OEIS A255487 counts free polyhypercubes
 OEIS A151514 counts onesided snake polyominoes
 OEIS A002013 counts free snake polyominoes
 OEIS A359068 counts onesided strip polyominoes
 OEIS A333313 counts free strip polyominoes
Table of results (2D)
Here are my results for the 2D polyomino cases (source code). Columns corresponding to existing OEIS sequences are marked accordingly. OEIS sequences that did not exist prior to this blog post are italicized.
n  Free strip polyominoes (A333313)  Free nonouroboros snakes (A002013)  Free ouroboroi (A359706)  Onesided strips (A359068)  Onesided nonouroboros snakes (A151514)  Onesided ouroboroi (A359707) 

1  1  1  1  1  
2  1  1  1  1  
3  2  2  2  2  
4  3  3  1  5  5  1 
5  7  7  10  10  
6  13  13  0  24  24  0 
7  30  31  52  53  
8  64  65  1  124  126  1 
9  150  154  282  289  
10  338  347  1  668  686  1 
11  794  824  1548  1604  
12  1836  1905  4  3654  3792  4 
13  4313  4512  8533  8925  
14  10067  10546  7  20093  21051  11 
15  23621  24935  47033  49638  
16  55313  58476  31  110533  116858  45 
17  129647  138002  258807  275480  
18  303720  323894  95  607227  647573  178 
19  711078  763172  1421055  1525113  
20  1665037  1790585  420  3329585  3580673  762 
21  3894282  4213061  7785995  8423334  
22  9111343  9878541  1682  18221563  19755938  3309 
23  21290577  23214728  42575336  46422915  
24  49770844  54393063  7544  99539106  108783480  14725 
25  116206114  127687369  232398659  255359883  
26  271435025  298969219  33288  542864111  597932342  66323 
27  633298969  701171557  1266567155  1402308318  
28  1478178004  1640683309  152022  2956342341  3281352516  302342 
29  3446626028  3844724417  6893180336  7689369625  
30  8039424324  8991137036  696096  16078817198  17982241557  1391008 
31  18734704111  21054243655  37469245219  42108302007  
32  43673728357  49211076053  3231001  87347384305  98422076879  6453950 
33  101723730306  115161584232  203447081205  230322745835 
Table of results (3D)
Here’s a partial table of 3D polycube cases. Columns corresponding to existing OEIS sequences are marked accordingly; which is to say, none of these sequences seem to be in the OEIS quite yet. Blanks (except in the odd ouroboros cases) indicate that I just haven’t computed those values yet; but I’m working on filling those in (source code).
The table below indicates the existence of two chiral pairs among the 10cube ouroboroi. Can you find them? How about the three different ways to enclose a cavity with a 12cube ouroboros (one mirrorsymmetric, the other two forming a chiral pair)?
n  Free nonouroboros snakes  Free nonouroboros snakes with cavities  Free ouroboroi  Free ouroboroi with cavities  Onesided nonouroboros snakes  Onesided nonouroboros snakes with cavities  Onesided ouroboroi  Onesided ouroboroi with cavities 
1  1  1  
2  1  1  
3  2  2  
4  4  1  5  1  
5  12  16  
6  34  1  54  1  
7  125  212  
8  450  3  827  3  
9  1780  3369  
10  7021  11  13653  13  
11  28521  4  56052  8  
12  115553  5  77  2  229004  10  122  3 
13  472578  24  939935  48  
14  1927634  105  606  0  3843859  210  1115  0 
15  7890893  485  15753903  970  
16  32221475  2098  6465  4  64380796  4196  12562  8 
17  131812746  9381  263475472  18762  
18  538059836  40566  74314  23  1075780425  81132  147350  46 
19  2198986587  178329  4397161320  356658  
20  8970624060  768163  907495  273  17939394036  1536321  1810165  545 
21  36628143111  3334895  73251877235  6669790  
22  149328243327  14303573  11415061  2980  298646347226  28607136  22812552  5960 
23  1218453344740 
See also:
 “Lego polycube snakes” (20221211)